Finite Group Databasebeta

← C23×#2778 #15670397

Subgroups — C23×#2778 #15670397

Order 2944.

498 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#401C211yesmaximal, characteristic, complemented
2C23×#400C211yesmaximal, characteristic, complemented
3C23×#397C211yesmaximal, characteristic, complemented
4C23×#394C211yesmaximal, characteristic, complemented
5C23×#394C211yesmaximal, characteristic, complemented
6C23×#392C211yesmaximal, characteristic
7C23×#388C211yesmaximal, characteristic, complemented
8C23×#388C211yesmaximal, characteristic, complemented
9C23×#388C211yesmaximal, characteristic, complemented
10C23×#386C211yesmaximal, characteristic, complemented
11C23×#386C211yesmaximal, characteristic, complemented
12C23×#386C211yesmaximal, characteristic, complemented
13C23×#386C211yesmaximal, characteristic, complemented
14C23×#385C211yesmaximal, characteristic, complemented
15C23×#380C211yesmaximal, characteristic, complemented
16C22×D4×C23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic
18C22×D4×C23C2211yescharacteristic
19C23×C4×C23C2211yescharacteristic
20C23×C4×C23C2211yescharacteristic
21C2×C4⋊C4×C23C2211yescharacteristic
22C2×C4⋊C4×C23C2211yescharacteristic
23C2×C4⋊C4×C23C2211yescharacteristic
24C2×C4⋊C4×C23C2211yescharacteristic
25C2×C22⋊C4×C23C2211yescharacteristic
26C2×C22⋊C4×C23C2211yescharacteristic
27C2×C22⋊C4×C23C2211yescharacteristic
28C2×C22⋊C4×C23C2211yescharacteristic
29C2×C22⋊C4×C23C2211yescharacteristic
30C2×C22⋊C4×C23C2211yescharacteristic
31C2×C22⋊C4×C23C2211yescharacteristic
32C2×C22⋊C4×C23C2211yescharacteristic
33C2×C22⋊C4×C23C2211yescharacteristic
34C2×C22⋊C4×C23C2211yescharacteristic
35C2×C22⋊C4×C23C2211yescharacteristic
36C2×C22⋊C4×C23C2211yescharacteristic
37C2×C22⋊C4×C23C2211yescharacteristic
38C2×C42×C23C2211yescharacteristic
39C23×#95C2211yescharacteristic
40C23×#95C2211yescharacteristic
41C23×#95C2211yescharacteristic
42C23×#95C2211yescharacteristic
43C23×#95C2211yescharacteristic
44C23×#95C2211yescharacteristic
45C23×#95C2211yescharacteristic
46C23×#95C2211yescharacteristic
47C23×#95C2211yescharacteristic
48C23×#95C2211yescharacteristic
49C23×#95C2211yescharacteristic
50C23×#95C2211yescharacteristic
51C24×C23C2311yescharacteristic
52C24×C23C2311yescharacteristic
53C24×C23C2311yescharacteristic
54C2×D4×C23—22no—
55C2×D4×C23—22no—
56C2×D4×C23—22no—
57C2×D4×C23—22no—
58C2×D4×C23—41no—
59C2×D4×C23—41no—
60C2×D4×C23—41no—
61C2×D4×C23—41no—
62C2×D4×C23—41no—
63C22×C4×C23—22no—
64C22×C4×C23—41no—
65C22×C4×C23—41no—
66C22×C4×C23—41no—
67C22×C4×C23—41no—
68C22×C4×C23—41no—
69C22×C4×C23C2311yescharacteristic
70C22×C4×C23C2311yescharacteristic
71C22×C4×C23C2311yescharacteristic
72C22×C4×C23C2311yescharacteristic
73C22×C4×C23C2311yescharacteristic
74C22×C4×C23C2311yescharacteristic
75C22×C4×C23C2311yescharacteristic
76C22×C4×C23C2311yescharacteristic
77C22×C4×C23C2311yescharacteristic
78C22×C4×C23C2311yescharacteristic
79C22×C4×C23C2311yescharacteristic
80C22×C4×C23C2311yescharacteristic
81C4⋊C4×C23—41no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—41no—
85C22⋊C4×C23—22no—
86C22⋊C4×C23—22no—
87C22⋊C4×C23—22no—
88C22⋊C4×C23—22no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C22⋊C4×C23—41no—
92C22⋊C4×C23—41no—
93C22⋊C4×C23—41no—
94C22⋊C4×C23—41no—
95C22⋊C4×C23—41no—
96C22⋊C4×C23—41no—
97C22⋊C4×C23—41no—
98C42×C23—41no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—21no—
105C23×C23—21no—
106C23×C23—21no—
107C23×C23—21no—
108C23×C23—21no—
109C23×C23—21no—
110C23×C23—21no—
111C23×C23—21no—
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23#4112yes—
119C23×C23#4112yes—
120C23×C23C2411yescharacteristic
121D4×C23—44no—
122D4×C23—44no—
123D4×C23—44no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—44no—
160C2×C4×C23—44no—
161C2×C4×C23#4112yes—
162#2778C2311yesmaximal, characteristic, direct product factor, Sylow
163C22×C23—22no—
164C22×C23—22no—
165C22×C23—22no—
166C22×C23—22no—
167C22×C23—22no—
168C22×C23—22no—
169C22×C23—41no—
170C22×C23—41no—
171C22×C23—41no—
172C22×C23—41no—
173C22×C23—41no—
174C22×C23—41no—
175C22×C23—41no—
176C22×C23—41no—
177C22×C23—41no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—41no—
184C22×C23C2×#4111yescharacteristic
185C22×C23C2×#4111yescharacteristic
186C22×C23C2×#4111yescharacteristic
187C22×C23H(5,𝔽2)11yescharacteristic
188C22×C23H(5,𝔽2)11yescharacteristic
189C22×C23H(5,𝔽2)11yescharacteristic
190C22×C23H(5,𝔽2)11yescharacteristic
191C4×C23—22no—
192C4×C23—41no—
193C4×C23—41no—
194C4×C23—41no—
195C4×C23—41no—
196C4×C23—41no—
197C4×C23—41no—
198C4×C23—41no—
199C4×C23—41no—
200C4×C23—41no—
201C4×C23—41no—
202C4×C23—41no—
203#401C2×C2311yescharacteristic, complemented
204#400C2×C2311yescharacteristic, complemented
205#397C2×C2311yescharacteristic, complemented
206#394C2×C2311yescharacteristic, complemented
207#394C2×C2311yescharacteristic, complemented
208#392C2×C2311yescharacteristic
209#388C2×C2311yescharacteristic, complemented
210#388C2×C2311yescharacteristic, complemented
211#388C2×C2311yescharacteristic, complemented
212#386C2×C2311yescharacteristic, complemented
213#386C2×C2311yescharacteristic, complemented
214#386C2×C2311yescharacteristic, complemented
215#386C2×C2311yescharacteristic, complemented
216#385C2×C2311yescharacteristic, complemented
217#380C2×C2311yescharacteristic, complemented
218C2×C23—42no—
219C2×C23—42no—
220C2×C23—81no—
221C2×C23#53811yescharacteristic
222C2×C23#53811yescharacteristic
223C2×C23#53811yescharacteristic
224C2×C23#55111yescharacteristic
225C2×C23#55311yescharacteristic
226C2×C23#55311yescharacteristic
227C2×C23#56011yescharacteristic
228C22×D4C22×C2311yescharacteristic
229C22×D4C22×C2311yescharacteristic
230C22×D4C22×C2311yescharacteristic
231C23×C4C22×C2311yescharacteristic
232C23×C4C22×C2311yescharacteristic
233C2×C4⋊C4C22×C2311yescharacteristic
234C2×C4⋊C4C22×C2311yescharacteristic
235C2×C4⋊C4C22×C2311yescharacteristic
236C2×C4⋊C4C22×C2311yescharacteristic
237C2×C22⋊C4C22×C2311yescharacteristic
238C2×C22⋊C4C22×C2311yescharacteristic
239C2×C22⋊C4C22×C2311yescharacteristic
240C2×C22⋊C4C22×C2311yescharacteristic
241C2×C22⋊C4C22×C2311yescharacteristic
242C2×C22⋊C4C22×C2311yescharacteristic
243C2×C22⋊C4C22×C2311yescharacteristic
244C2×C22⋊C4C22×C2311yescharacteristic
245C2×C22⋊C4C22×C2311yescharacteristic
246C2×C22⋊C4C22×C2311yescharacteristic
247C2×C22⋊C4C22×C2311yescharacteristic
248C2×C22⋊C4C22×C2311yescharacteristic
249C2×C22⋊C4C22×C2311yescharacteristic
250C2×C42C22×C2311yescharacteristic
251#95C22×C2311yescharacteristic
252#95C22×C2311yescharacteristic
253#95C22×C2311yescharacteristic
254#95C22×C2311yescharacteristic
255#95C22×C2311yescharacteristic
256#95C22×C2311yescharacteristic
257#95C22×C2311yescharacteristic
258#95C22×C2311yescharacteristic
259#95C22×C2311yescharacteristic
260#95C22×C2311yescharacteristic
261#95C22×C2311yescharacteristic
262#95C22×C2311yescharacteristic
263C23#277811yescharacteristic, direct product factor, Sylow
264C24C23×C2311yescharacteristic
265C24C23×C2311yescharacteristic
266C24C23×C2311yescharacteristic
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4C23×C2311yescharacteristic
283C22×C4C23×C2311yescharacteristic
284C22×C4C23×C2311yescharacteristic
285C22×C4C23×C2311yescharacteristic
286C22×C4C23×C2311yescharacteristic
287C22×C4C23×C2311yescharacteristic
288C22×C4C23×C2311yescharacteristic
289C22×C4C23×C2311yescharacteristic
290C22×C4C23×C2311yescharacteristic
291C22×C4C23×C2311yescharacteristic
292C22×C4C23×C2311yescharacteristic
293C22×C4C23×C2311yescharacteristic
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—41no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23#41×C2312yes—
332C23#41×C2312yes—
333C23C24×C2311yescharacteristic
334D4—44no—
335D4—44no—
336D4—44no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4#41×C2312yes—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22C2×#41×C2311yescharacteristic
397C22C2×#41×C2311yescharacteristic
398C22C2×#41×C2311yescharacteristic
399C22C23×H(5,𝔽2)11yescharacteristic
400C22C23×H(5,𝔽2)11yescharacteristic
401C22C23×H(5,𝔽2)11yescharacteristic
402C22C23×H(5,𝔽2)11yescharacteristic
403C4—22no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C2—42no—
416C2—42no—
417C2—81no—
418C2C23×#53811yescharacteristic
419C2C23×#53811yescharacteristic
420C2C23×#53811yescharacteristic
421C2C23×#55111yescharacteristic
422C2C23×#55311yescharacteristic
423C2C23×#55311yescharacteristic
424C2C23×#56011yescharacteristic
subgroupG/H
center, socleC23×C23C24
commutator, Frattini, cosocleC23C24×C23
prime radicalC23×#392C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×#401C2semidirect
2. C23×#400C2semidirect
3. C23×#397C2semidirect
4. C23×#394C2semidirect
5. C23×#394C2semidirect
7. C23×#388C2semidirect
8. C23×#388C2semidirect
9. C23×#388C2semidirect
10. C23×#386C2semidirect
11. C23×#386C2semidirect
12. C23×#386C2semidirect
13. C23×#386C2semidirect
14. C23×#385C2semidirect
15. C23×#380C2semidirect
162. #2778C23direct (N × H)
203. #401C2×C23semidirect
204. #400C2×C23semidirect
205. #397C2×C23semidirect
206. #394C2×C23semidirect
207. #394C2×C23semidirect
209. #388C2×C23semidirect
210. #388C2×C23semidirect
211. #388C2×C23semidirect
212. #386C2×C23semidirect
213. #386C2×C23semidirect
214. #386C2×C23semidirect
215. #386C2×C23semidirect
216. #385C2×C23semidirect
217. #380C2×C23semidirect
263. C23#2778direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#4012
C23×#4002
C23×#3972
C23×#3942
C23×#3922
C23×#3882
C23×#3862
C23×#3852
C23×#3802
#277823